Suppose that to every non-degenerate simplex
$\Delta\subset{\mathbb{R}}^{n}$ a ‘center’ $C(\Delta)$ is assigned so that the
following assumptions hold:

(i)

The map $\Delta\to C(\Delta)$ commutes with similarities and is
invariant under the permutations of the vertices of the simplex;

(ii)

The map $\Delta\to\mathrm{Vol}(\Delta)C(\Delta)$ is polynomial in the coordinates of
the vertices of the simplex.

Then $C(\Delta)$ is an affine combination of the center of mass
$CM(\Delta)$ and the circumcenter $CC(\Delta)$ of the simplex:

$C(\Delta)=tCM(\Delta)+(1-t)CC(\Delta),$

where the constant $t\in{\mathbb{R}}$ depends on the map $\Delta\mapsto C(\Delta)$ (and does not depend on the simplex $\Delta$). The
motivation for this theorem comes from the recent study of the
circumcenter of mass of simplicial polytopes by the authors and by
A. Akopyan.

Given a homogeneous polygonal lamina $P$, one way to find its center
of mass is as follows: triangulate $P$, assign to each triangle its
centroid, taken with the weight equal to the area of the triangle,
and find the center of mass of the resulting system of point masses.
That the resulting point, $CM(P)$, does not depend on the
triangulation, is a consequence of Archimedes’ Lemma: if an
object is divided into smaller objects, then the center of mass of
the compound object is the weighted average of the centers of mass
of the parts, with the weights equal to the respective areas.

Replace, in the above construction, the centroids of the triangles
by their circumcenters. The resulting weighted average is called the
circumcenter of mass of the polygon $P$, denoted by $CCM(P)$.
This point is well defined, that is, does not depend on the
triangulation (assuming that degenerate triangles are avoided), see
Fig. 1.

Figure 1: Circumcenter of mass

This construction is mentioned in the 19th century book
([Laisant1887]),
where it is attributed to the Italian algebraic
geometer G. Bellavitis. We learned about this reference from B.
Grünbaum who, together with G. C. Shephard, studied this
at about the same time, the circumcenter of mass was rediscovered by
[Adler1993];
[Adler1995]
as an integral of a discrete dynamical system called
recutting of polygons.

The explicit formulas are as follows. Let the coordinates of the
vertices of the polygon $P$, taken in the cyclic order, be
$(x_{i},y_{i}),\ i=1,\ldots,n$. Then $CCM(P)=\frac{1}{4A(P)}\left(\sum_{i=0}^{n-1}y_{i}(x_{i-1}^{2}+y_{i-1}^{2}-x_{i+1}^{2}
-y_{i+1}^{2}),\sum_{i=0}^{n-1}-x_{i}(x_{i-1}^{2}+y_{i-1}^{2}-x_{i+1}^{2}-y_{i+
1}^{2})\right),$ where $A(P)$ is the signed area of $P$ (see
[Tabachnikov and Tsukerman2014]
for a proof).
For comparison, $CM(P)=\frac{1}{6A(P)}\left(\sum_{i=0}^{n-1}(x_{i}+x_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}
),\sum_{i=0}^{n-1}(y_{i}+y_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i})\right)$, a result of a straightforward calculation.

The construction of the circumcenter of mass extends to higher
dimensions, and to the elliptic and hyperbolic geometries. We
studied it in
[Tabachnikov and Tsukerman2014]
in relation with the so-called discrete
bicycle transformation ([Tabachnikov and Tsukerman2013]).
See also the paper by
[Akopyan2014]
.

The construction in ${\mathbb{R}}^{n}$ is similar. Given a simplicial polytope
$P$, consider its non-degenerate triangulation. Assign the
circumcenter $CC(\Delta_{i})$ to each simplex $\Delta_{i}$ of the
triangulation, and take the center of mass of these points with
weights equal to the oriented volumes of the respective simplices:

The explicit formula is as follows. Let $F=(V_{1},\ldots,V_{n})$ be a
face of $P$, where $V_{i}$ are vectors in ${\mathbb{R}}^{n}$. Let $A(F)$ be the
$n\times n$ matrix made of vectors $V_{i}$, and let $A_{i}(F)$ be
obtained from $A(F)$ by replacing $i$th row with
$(|V_{1}|^{2},\ldots,|V_{n}|^{2})$. Then the $i$th component of the
circumcenter of mass is given by

In the above formulas, the signs of $\mathrm{Vol}(\Delta_{i})$ and the orders
of vertices $V_{1},\ldots,V_{n}$ are chosen consistently: we consider
the triangulation as a simplicial chain.

One can take affine combinations $tCM+(1-t)CCM,\ t\in{\mathbb{R}}$,
resulting in a line, called the generalized Euler line of the
polytope $P$ (for a triangle, the Euler line is the line through the
centroid and the circumcenter; it passes through the orthocenter as
well).

In this note we are interested in the uniqueness of this
construction.

Suppose that to every non-degenerate simplex $\Delta\subset{\mathbb{R}}^{n}$ a
‘center’ $C(\Delta)\in{\mathbb{R}}^{n}$ is assigned so that the following
assumptions hold:

Let $x_{1},\ldots,x_{n}$ be Cartesian coordinates in ${\mathbb{R}}^{n}$. Let $\Delta=(V_{0},\ldots,V_{n})$ be a simplex, and let $V_{j}=(x_{1}^{j},\ldots,x_{n}^{j}),\ j=0,\ldots,n$, be the coordinates of its vertices (where $j$ is a
superscript, not an exponent). Let

where Skew is skew-symmetrization over superscripts. Evidently,
$X_{i,jk}=X_{i,kj}$, and the number of such polynomials equals
$n^{2}(n+1)/2$. Both determinants, $V$ and $X_{i,jk}$, are
skew-symmetric under permutations of the vertices of the simplex.

Lemma 2.1.

The polynomials $X_{i,jk}$ constitute a linear basis of the space
${\cal S}$ of homogeneous polynomials of degree $n+1$ in the
variables $x_{1}^{0},x_{2}^{0},\ldots,x_{n}^{n}$, skew-symmetric under
permutations of the superscripts.

The superscripts $i-1$ and $\sigma(i^{\prime}-1)$ are the unique ones which
occur twice. Therefore $\sigma(i^{\prime}-1)=i-1$. The corresponding
subscripts are $j,k$ and $j^{\prime},k^{\prime}$, so that $\{j,k\}=\{j^{\prime},k^{\prime}\}$.
Dividing both sides by $x_{j}^{i-1}x_{k}^{i-1}$, we get

Consider the skew-symmetrization of a monomial $x_{\beta}^{\alpha}$.
If some number appears in $\alpha$ with multiplicity 3 or greater,
then $\alpha$ must be missing some two distinct numbers $i,j\in\{0,1,\ldots,n\}$. Each permutation $\sigma$ has a counterpart
$\sigma(i\,j)$ of opposite sign which maps $x_{\beta}^{\alpha}$ to
the same monomial. Therefore the skew-symmetrization of
$x_{\beta}^{\alpha}$ in this case is zero.

Now suppose that $\alpha$ contains $n+1$ different elements of
$\{0,1,\ldots,n\}$. Since the entries of $\beta$ are elements of
$\{1,2,\ldots,n\}$, there exist some $\beta_{i}$ and $\beta_{j}$ with $i\neq j$ such that $\beta_{i}=\beta_{j}$. Each permutation $\sigma$ has a
counterpart $\sigma(\alpha_{i}\,\alpha_{j})$ of opposite sign which
maps $x_{\beta}^{\alpha}$ to the same element.

It follows that the only monomials appearing in $f$ are those for
which $\alpha$ is a permutation of
$(0,1,\ldots,\widehat{i-1},\ldots,n-1,i-1,i-1)$. Assume without loss
of generality that $\alpha$ is of this form. Let
$\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n},\alpha_{n+1})$ and
$\beta=(\beta_{1},\beta_{2},\ldots,\beta_{n},\beta_{n+1})$. Suppose that
$\beta_{i_{1}}=\beta_{i_{2}}=\ldots=\beta_{i_{k}}$. For the monomial to
not vanish under skew-symmetrization, the corresponding multiset
$\alpha_{i_{1}},\alpha_{i_{2}},\ldots,\alpha_{i_{k}}$ cannot be invariant
under any transpositions. Knowing the structure of $\alpha$, we see
that this implies that $k=1,2$ or 3. If $k=2$, then
$(\alpha_{i_{1}},\alpha_{i_{2}})=(i-1,i-1)$. If $k=3$ then
$(\alpha_{i_{1}},\alpha_{i_{2}},\alpha_{i_{3}})=(i-1,i-1,q)$, with $q\neq i-1$. This proves the claim. $\square$

Consider the map $\varphi:\Delta\mapsto\mathrm{Vol}(\Delta)C(\Delta)$,
and let $(y_{1},\ldots,y_{n})$ be its components. Our assumption 3
implies that each $y_{\ell},\ \ell=1,\ldots,n$, is a polynomial in the
variables $x_{1}^{0},x_{2}^{0},\ldots,x_{n}^{n}$. The assumption 1, applied to
scaling, implies that these polynomials are homogeneous of degree
$n+1$, and the assumption 2 that they are skew-symmetric under
permutations of the superscripts. Lemma 2.1 implies that

$y_{\ell}=\sum A_{i,jk}^{\ell}X_{i,jk},\quad\ell=1,\ldots,n,$

where the coefficients $A_{i,jk}^{\ell}$ satisfy $A_{i,jk}^{\ell}=A_{i,kj}^{\ell}$. We always assume that summation is over repeated
indices.

Example 2.2.

The center of mass and the circumcenter of mass correspond to the
functions

Then, taking the orientations of the simplices into account, the
above equality for determinants yields the result.
$\square$

Lemma 2.3 is a particular case of Archimedes’ Lemma: the
triangulation is conical, obtained by connecting the origin to the
vertices of the simplex. It is shown in
[Tabachnikov and Tsukerman2014]
that this
particular conical case implies Archimedes’ Lemma for all
triangulations without extra points on the boundary [see
Sect. 4(i) for a discussion].

In the next section, we shall use assumption 1, namely, the fact
that the map $\Delta\mapsto C(\Delta)$ commutes with parallel
translations, rotations, and transpositions of coordinates, to
conclude that the coefficients $A_{i,jk}^{\ell}$ must be affine
combinations of the ones in (2).

A transposition of coordinates is reflection in a hyperplane, and it
changes the sign of $V$. According to Lemma 3.1, a
transposition also changes the sign of the basic polynomials:
$X_{i,jk}\mapsto-X_{\sigma(i),\sigma(j)\sigma(k)}$. Hence the
covariance of the map $\Delta\mapsto C(\Delta)$ with respect to
$\sigma$ implies the equality

On the other hand, $y_{\ell}/V$ is the $\ell$th component of the map
$\Delta\mapsto C(\Delta)$, and the infinitesimal translation in the
$r$th direction sends it to $\delta_{\ell r}$. By translation
covariance, the above sum equals $\delta_{\ell r}$, as claimed.

Likewise, the infinitesimal rotation in the $p,q$-plane annihilates
$y_{\ell}$ for $\ell$ distinct from $p,q$, and sends $y_{q}$ to $y_{p}$,
and $y_{p}$ to $-y_{q}$; in short,

and then, for fixed $a,b,c$, equate the coefficients in front of
$X_{a,bc}$ in both expressions to obtain (5). $\square$

Now we need to solve the system of linear Eqs.
(3) – (5) on the unknowns $A_{i,jk}^{\ell}$. We use
(3) to reduce the number of variables.

Consider the following four cases. If $|\{i,j,k,\ell\}|=4$ then,
applying an appropriate sequence of transpositions, we obtain:
$A_{i,jk}^{\ell}=A^{1}_{2,34}=:t$. If $|\{i,j,k,\ell\}|=3$, then one
has four sub-cases, and $A_{i,jk}^{\ell}$ is equal to

and (5) $\ell=q=c=b$, but distinct from pairwise distinct
$p,a$, yields $2v+w=0.$ We have obtained four linear equations on
$u,v,w,\alpha$, and the only solution of this system is zero. This
completes the proof.

(i) Degenerate simplices can be safely ignored when calculating the
center of mass: such a simplex has a finite centroid and zero
volume, making no contribution to the total sum. Not so for the
circumcenter of mass: although the volume of a nearly degenerate
simplex tends to zero, its circumcenter may go to infinity, and the
contribution to the sum (1) may be non-negligible. The
map $\varphi:\Delta\mapsto\mathrm{Vol}(\Delta)\ CC(\Delta)$, being
polynomial in the coordinates of the vertices, is continuous.

For example, consider an isosceles right triangle $ABC$. Its
circumcenter is the midpoint $M$ of the hypothenuse $AC$. Consider
the triangulation in Fig. 2 consisting of three
triangles, one of which, $AMC$, is degenerate. If one ignored this
triangle, then, by Archimedes’ Lemma, the circumcenter of mass of
$\triangle ABC$ would be the midpoint of the segment connecting the
midpoints of the hypothenuses $AB$ and $BC$ of the triangles $ABM$
and $BCM$. The latter point is the circumcenter of the quadrilateral $ABCM$, not the triangle $ABC$.

Figure 2: Contribution of a degenerate triangle

(ii) One may wish to extend the notion of the circumcenter of mass
to more general sets. For example, let $\gamma(t)$ be a
parameterized smooth curve, star-shaped with respect to point $O$,
see Fig. 2. It is natural to define the circumcenter of
mass by continuity as

$\frac{\int{C}(t)\ dA}{\int dA},$

(7)

where $C(t)$ denotes the limiting $\varepsilon\to 0$ position of
the vector from $O$ to the circumcenter of the infinitesimal
triangle $O\gamma(t)\gamma(t+\varepsilon)$, and $dA$ is the area
of this infinitesimal triangle. However, this does not give
anything new: the integral (7) is the center of mass of
the lamina bounded by the curve
([Tabachnikov and Tsukerman2014]).

Figure 3: Continuous limit of the circumcenter of
mass

(iii) Although the rational map $\Delta\mapsto CC(\Delta)$ is
discontinuous, the polynomial map $\varphi:P\mapsto\mathrm{Vol}(P)\ CCM(P)$, defined on simplicial polytopes in ${\mathbb{R}}^{n}$, is continuous
and is a valuation:

This valuation is isometry covariant; see, e.g.,
[Schneider2014]
for the
theory of valuations.

Continuity is with respect to the following topology in the space of
polytopes in ${\mathbb{R}}^{n}$: if the vertices of a polytope $P$ are
$V_{1},\ldots,V_{k}$, then we view $P$ as a point in ${\mathbb{R}}^{n}\times\ldots\times{\mathbb{R}}^{n}$ ($k$ times), and the topology is that of
${\mathbb{R}}^{nk}$.

(iv) We finish with open problems. Consider the same topology on the
space of polytopes as in (iii).

Problem 4.1.

As we mentioned, the construction of the circumcenter of mass
extends to the spherical and hyperbolic geometries
([Akopyan2014];
[Tabachnikov and Tsukerman2014]). It
is interesting to find an axiomatic description of the centers for
simplicial polygons and polytopes, discussed in this note, in the
three geometries of constant curvature (for the center of mass, see
[Galperin1993]). These centers should be isometry covariant and
satisfy some additivity condition (Archimedes’ Lemma or
valuation-like). Such a description should include the valuations
from Problem4.1. At the moment of writing, we do not know
such an axiomatic description.

This work is an extension of the project that originated
in the Summer@ICERM 2012 program; it is a pleasure to acknowledge
the inspiring atmosphere and hospitality of the institute. We are
grateful to V. Adler, A. Akopyan, Yu. Baryshnikov and B. Grünbaum
for their interest and help. Many thanks to the referees for their
suggestions and criticism. The first author was supported by the
NSF grant DMS-1105442, and the second author by a NSF Graduate
Research Fellowship under Grant No. DGE 1106400.